Venn Diagrams (Cambridge International AS Mathematics)

Venn Diagrams

What is a Venn diagram?

• A Venn diagram is a way to illustrate events and are particularly useful when outcomes overlap

• Venn diagrams are mostly used for 2 or 3 events

• A Venn diagram consists of a box (rectangle) and a bubble (circle/ellipse) for each event

  • Bubbles may or may not overlap
  • Bubble(s) is not a technical term, but we like it!

• The box represents all outcomes

  • It is often referred to as the Universal Set and is commonly labelled with the symbols $E$ (Kunstler script font uppercase letter E)$S,U\mathrm{or}ξ$ (Greek lowercase letter Xi)
  • There is no standardised symbol for this purpose

• Bubbles are labelled with their event name (A, B, etc)

• The numbers inside a Venn diagram (there should be one in each region) will represent either a frequency or a probability

  • In the case of probabilities being shown, all values should total 1

What do the different regions and bubbles overlapping mean on a Venn diagram?

• This will depend on how many events there are and how the outcomes overlap
• Venn diagrams show ‘AND’ and ‘OR’ statements easily

Worked Example 40 people were surveyed regarding which games consoles they owned.

8 people said they owned a Playstation 5 ($P$) and an Xbox Series X ($X$).

5 people said they owned neither of these consoles.

Of those people that owned only one games console, twice as many owned an Xbox Series X as a Playstation 5.

One of the 40 people is chosen at random. Find the probability that this person

(i) owns both consoles,

(ii) owns exactly one console,

(iii) doesn’t own a Playstation 5.

Answer:

Exam Hint

• The rectangle in a Venn diagram is a key part of the diagram

  • it represents all possible outcomes of the experiment
  • the bubbles merely represent the events we are particularly interested in
  • there is usually a few possibilities that fall outside of these events so this would be the section outside the bubbles but inside the rectangle

• A quick ‘mini-Venn’ diagram shading the parts required to answer the question can be useful rather than always drawing a full Venn diagram with all its values